How to move 3D cursor to point in bezier circle?
This is probably more general than you need:

To establish a point on the circumference of a curve circle at a known angle from its local X axis.

Shortcuts for 2.79



Assuming a circle at an arbitrary location and rotation:


Select the circle. ShiftNumpad 7 align the view to it, and ShiftS put the 3D cursor to its center. 
CtrlAltSpace create a custom orientation from the circle; Check 'Use After Creation'. ShiftA create a Mesh: Single Vertex. EXX the vertex outward in X of the circle's space, with snap set to 'Vertex','Active'.  There should be a control point to snap to. 
RZZ the extruded vertex through the known angle, with '3D Cursor' as pivot.(Illustrated with a duplicate edge)
With the vertex selected, ShiftS snap the 3D cursor to it.


Your circle is probably already aligned to the world, so you may be able to skip the Custom Orientation steps.

In an even worse situation, though, where the circle's own control points are not aligned with its local axes (with Custom Orientation and Snap set as before):


Extrude a vertex to any available control point on the circle to establish a radius
Extrude another vertex from the circle's center, in the X of the circle's space (EXX) to any point outside the circle
Rotate RZZ the radius vertex about the center of the circle, with snap on, until the rotation snaps to the other vertex  you sent out in local X.